Propriedades estatísticas de bilhares abertos - Francisco, Matheus Hansen [UNESP] (2015)
Billiards are dynamical systems where a classical particle of mass m moves confined inside a boundary ∂Q to which suffers specular collisions. When the boundary is static, the kinetic energy of the particle is constant, hence its velocity. On the other hand, when a time perturbation is introduced in the boundary, depending on the phase of the moving wall as well the velocity, the particle can gain or lose energy upon collision. In this work, we study the oval billiard considering either the static as well as the time perturbation in the boundary. For the static boundary, the dynamics is described by a two dimensional, nonlinear mapping for the variables θ, corresponding to the polar angle and α denoting the angle the trajectory of the particle does with the tangent at the point of collision. We confirm the phase space is mixed containing both chaos, periodic islands as well as invariant spanning curves corresponding to the so called whispering gallery orbits. The chaotic sea is characterised via Lyapunov exponents. We concentrate particularly on the escape of particles from a hole in the boundary. We give convincing arguments the survival probability is described by an exponential function for short n and may change for a slower decay at larger n due to the stickiness phenomenon. The slope of the exponential decay scales with the relative size of the hole of the boundary. For the time dependent perturbation, the dynamics is described by a four dimensional and nonlinear mapping for the two previous angle variables plus the velocity of the particle and the time. The survival probability is also described by an exponential function for short n and, occasionally, a dynamical trapping produced by stickiness is observed too, therefore slowing down the speed of the decay of the survival probability